Loaded die whose one-roll and two-roll laws agree

A six-sided die showing $1$ to $6$ is loaded so that, for each $v\in\{2,3,4,5,6\}$, the chance of getting $v$ on a single roll equals the chance that the sum of two independent rolls equals $v$. Writing $p_1$ for the probability of a $1$, it turns out $p_1$ is the unique positive root of an equation $\sum_{k=1}^{6}c_k p_1^{\,k}=1$ with real coefficients $c_k$. What is $c_1+c_2+\dots+c_6$?

$65$. Matching two-roll sums gives $p_2=p_1^2$, $p_3=2p_1p_2$, $p_4=2p_1p_3+p_2^2$, and so on, so $p_k=C_{k-1}p_1^k$ with Catalan numbers $1,1,2,5,14,42$. Normalising, $\sum_k C_{k-1}p_1^k=1$, and the coefficients sum to $1+1+2+5+14+42=65$.